The field of the present invention relates to optical diffraction gratings. In particular, apparatus and methods are disclosed that include thermally compensated optical diffraction gratings, including both transmission gratings and reflection gratings.
A variety of undesirable or adverse operating characteristics of diffraction gratings arise when they are deployed in environments wherein temperature is not a constant. Many diffraction grating performance parameters, e.g., diffraction efficiency, diffraction angle of a given wavelength, or polarization-dependent diffraction efficiency, exhibit temperature dependence. Such temperature dependence can result in grating performance outside an operationally acceptable range. Temperature changes in the grating's use environment can affect grating performance parameters through temperature-induced changes of the refractive index of the surrounding medium or grating substrate, or through thermal expansion or contraction of the grating substrate material, or through variation of other parameters.
A conventional optical transmission grating 100 is illustrated schematically in FIG. 1. The grating 100 comprises a substrate 102 and a set of grating lines 104. The substrate 102 is substantially transparent over an operational wavelength range, has a temperature-dependent refractive index nsub, and has substantially flat, substantially parallel first and second surfaces. The grating is immersed in a substantially transparent ambient medium 110 having a temperature-dependent refractive index nmed (air in the example, with temperature-dependent refractive index nmed=nAir; other substantially transparent solid, liquid, or gaseous media can be employed). A cross section of the grating 100 is shown in FIG. 1, in which the grating is viewed in a direction parallel to the grating lines 104 and perpendicular to a plane of incidence defined by a surface normal vector and an incident wavevector kin. The set of grating lines 104 in the illustrated example are a set of parallel grooves etched into a first surface of the substrate 102 and evenly spaced by a grating period Λ (equivalently, the grating lines can be viewed as the set of parallel ridges protruding between the grooves). The grating period Λ is sometimes comparable to or shorter than the wavelength λ of an incident optical signal (e.g., between about 800 nm and about 2000 nm), so that only one diffracted order is present, as in the example shown. However, any suitable grating spacing can be employed, resulting in a corresponding number of one or more diffracted orders.
In the example shown, the grating lines comprise grooves etched in the substrate (or the ridges between them), and those grooves/ridges are shown as substantially rectangular in cross section. However, any suitable cross-sectional shape for grooves or ridges can be employed for the grating lines, e.g., rectangular, trapezoidal, sawtooth, triangular, blazed, or sinusoidal. Each grating line can itself comprise a set of multiple grooves, ridges, or other features. In the example shown, the grating teeth (i.e., the ridges protruding between the etched grooves) comprise the same material as the substrate 102. However, grating lines can be implemented by forming the grating teeth from a material differing from the substrate material. Alternatively, the grating lines need not comprise any deviation from a flat grating surface at all, but can instead comprise any suitable periodic index modulation of the substrate or one or more layers on the substrate, made by altering substrate or layer material or by applying or depositing differing material. Apparatus and methods disclosed herein are applicable regardless of the manner in which the grating lines are formed in or on the substrate.
In FIG. 1, an optical signal with wave vector kin is incident on the transmission grating 100 with incidence angle θin (with respect to a grating surface normal vector). The transmission grating 100 diffracts a transmitted portion of that incident signal as a diffracted signal (negative first order), which exits the transmission grating substrate 102 with wavevector kt-1 at diffraction angle θd. In FIG. 1, zeroth order signals (i.e., specularly reflected and directly transmitted waves), any other transmitted diffracted orders, and any reflected diffracted orders are omitted for clarity.
The diffraction angle θd of the incident signal with incidence angle θin and wavelength λ is given by Eq. 1:sin θin+sin θd=λ/(Λ·nAir).  (1)
The temperature dependence of the grating period Λ and of the refractive index of air are given approximately by Eqs. 2 and 3:Λ(T)≈Λ(T0)·(1+(dL/dT)·(T−T0)/L) and  (2)nAir(T)≈nAir(T0)+(dnAir/dT)·(T−T0),  (3)where (dL/dT)/L is the thermal expansion coefficient of the substrate and dnAir/dT is the thermo-optic coefficient of air. As seen clearly from Eq. 1, θd is a function of temperature by virtue of the temperature dependences of the grating period and the ambient medium's refractive index. FIG. 2 shows θd(T) over an operational temperature range of −5° C. to 65° C. for: Λ(T0=25° C.)=1063.75 nm (940 lines/mm); λ=1545 nm; =46.5°; a fused silica substrate (i.e., (dL/dT)/L=4.5×10−7); nAir(T0=25° C.)=1.0003; and dnAir/dT=−0.85×10−6. As can be seen in FIG. 2, the diffraction angle θd increases with increasing temperature, with a slope of about 0.00005° per ° C. (i.e., about 5×10−5 angular degrees per degree Celsius).
The variation of the diffraction angle θd with temperature is given more generally by Eq. 4:
                                                                                          ⅆ                                      θ                    d                                                                    ⅆ                  T                                            =                                                                    -                    λ                                                                              n                      med                                        ⁢                    Λ                    ⁢                                                                                  ⁢                    cos                    ⁢                                                                                  ⁢                                          θ                      d                                                                      ⁢                                  (                                                                                    1                        Λ                                            ⁢                                                                        ⅆ                          Λ                                                                          ⅆ                          T                                                                                      +                                                                  1                                                  n                          med                                                                    ⁢                                                                        ⅆ                                                      n                            med                                                                                                    ⅆ                          T                                                                                                      )                                                                                                        =                                                                    -                    λ                                                                              n                      med                                        ⁢                                          Λcosθ                      d                                                                      ⁢                                  (                                                            CTE                      sub                                        +                                                                  1                                                  n                          med                                                                    ⁢                                              TOC                        med                                                                              )                                                                                        (        4        )            where CTEsub is the coefficient of thermal expansion of the substrate and TOCmed is the thermo-optic coefficient of the ambient medium. Similar equations can be derived for other diffracted orders (transmitted or reflected).
Looking more closely at the relative influence on the diffraction angle θd of the two temperature-dependent quantities, air refractive index and grating period, the following is observed (at least for the example of a fused silica substrate in air). With increasing temperature, the grating period Λ increases due to the positive coefficient of thermal expansion (CTE) of the substrate material. Eq. 1 therefore implies that the diffraction angle θd would decrease in the absence of any temperature-dependent change of the refractive index of the ambient medium. But the refractive index of the ambient medium decreases with temperature due to the negative thermo-optic coefficient (TOC) of the ambient medium. Eq. 1 therefore implies that the diffraction angle θd would increase in the absence of any temperature-dependent substrate expansion. In this example (fused silica in air), the second effect outweighs the first, resulting in a diffraction angle θd that increases with temperature, as seen in FIG. 2. The temperature dependence of the substrate refractive index nsub does not influence the diffraction angle θd for the grating 100 having parallel first and second surfaces, as is the case in the example of FIGS. 1 and 2.
It should be noted that Eqs. 1-4 apply to an optical diffraction grating as shown in FIG. 1 whether it is used in transmission (as shown) or in reflection, and for either of those cases whether the incident optical signal is first incident on the surface having the grating lines (as shown) or on the other surface. In all of those cases, parallel first and second substrate surfaces result in no dependence of the diffraction angle on the substrate index nsub, and all of those gratings exhibit temperature-dependent diffractive behavior according to Eqs. 1-4.
It would be desirable to provide an optical diffraction grating exhibiting reduced temperature dependence of grating performance, including the diffraction angle θd.